CHAPTER TEN

CONDUCTING SCIENTIFIC INQUIRY

10.1 Types of Scientific Experiments 10.2 Generic Experimental Design 10.3 Making Measurements 10.4 Dealing with Uncertainties 10.5 The Nature of Experimentation 10.6 Planning an Experiment 10.7 Practices in the Laboratory 10.8 Evaluating an Experiment 10.9 Report Writing

Conducting scientific inquiry accurately is an acquired art form. While no “scientific method” per se exists, scientists frequently conduct different types of experiments among which there is a certain amount of commonality. This chapter takes a broad rather than detailed approach to conducting a scientific experiment. There is no doubt that only a full course in the methods of scientific inquiry could even begin to scratch the surface of all the details required to conduct a scientific experiment in the most professional manner. This chapter should, however, provide the physics teacher candidate or high school physics teacher with sufficient information needed to conduct this form of inquiry at an age-appropriate level.

Types of Scientific Experiments Observation Testing Application

Generic Experimental Design

While scientists will agree that there is no “scientific method” per se, there is basic agreement about what fundamental procedures can be followed for designing and conducting a physics experiment. For instance, consider the following steps:

1. Identify the problem to be solved. 2. Identify the system being studied. 3. Identify and distinguish system variables. 4. Identify the general procedure to be followed. 5. Identify the model if possible. 6. Choose the range for the variables. 7. Collect and interpret data. 8. Consider the overall precision of the experiment.

These steps, if nothing more, make it clear that a number of important factors need to be considered even for generic experimental design. Let’s examine each of these factors in turn using a generic example. Refer to the following diagram as appropriate.

ho

hidido

object

imagepinhole

.

.

Step 1. Identify the problem to be solved. A physicist wants to experimentally determine the relationships associated with image formation in a pinhole camera (see the image above). The physicist seeks to find all the relationships between the height of an object, ho, the height of its image, hi, the distance of the object, do, and the distance of the image, di, as measured from the pinhole. Step 2. Identify the system being studied. Just how the physicists solves the problem will depend upon the system being studied. Clearly, this problem deals with pinhole projection, so the system will consist of a light bulb (the object), a pinhole in a screen, and a screen where the image will form. Step 3.Identify and distinguish system variables. Four of the system variables have already been clearly identified: hi, ho, di, and do. Are these the only system variables? No. There are other things that might vary in this experiment such as the size of the pinhole, the shape of the pinhole, the number of pinholes, the materials out of which the system is made, the kind of light bulb used, the nature of the material in which the pinhole is made and so on. Some variables are pertinent (e.g., hi, ho, di, and do), some are extraneous (e.g., probably the materials of which the system is made, the kind of light bulb used, the nature of the material in which the pinhole is made). There are other variables that are pertinent and must be controlled (e.g., the shape, size, and number of the pinholes). Other variables will be allowed to change and will be identified as either dependent or independent variables. Controlled variables are often known as system parameters. Step 4. Identify the general procedure to be followed. Scientists conduct controlled experiments. In controlled experiments there will be only one independent and one dependent variable at any one time. Other variables are held constant and are considered during that phase of experimentation to be state variables. The reason we have only one independent and one dependent variable at a time is so that we can determine the unique relationship between these two variables. If two or more variables are allowed to change independent, there is no way of telling how much effect each of the independent variables has on the dependent variable. Clearly, in our experiment we need to allow four variables to change during different phases of the experiment, but this must be done in such a way that there is only one independent and one dependent variable at a time. The other pertinent variables will be held constant. To begin the experimental study, the physicist decides to see what how varying di affects hi. During this phase of the experiment do and ho are held constant. During additional phases of the experiment there will be other combinations of hi, ho, di, and do always with one independent, one dependent and two controlled variables. Step 5. Identify the model if possible. Some times a theoretical analysis of the system will point to expected outcomes. This analysis can help interpret the data. In our hypothetical experiment, the physicist relies upon knowledge of the straight-line propagation of light and geometry to predict the outcome of the experiment. The real problem then is to determine whether or not experimental evidence supports the theoretical model. Sometimes it is not possible to directly predict the relationship being studied using theoretical approach, and such pragmatic approaches as dimensional analysis may be used to at least get a general understanding of what to expect. Step 6. Choose the range for the variables. This is one of the most important considerations for a number of reasons. First, collecting only a small range of data it might be impossible to distinguish between, say, a linear model, and inverse model, and a power function model. Even a parabolic or hyperbolic function appear linear when a small portion of the curve is considered. Generally speaking, it is best to have as wide a range of data as possible. Determine the extrema, and then consider determining an appropriate number of points between the extrema at which to collect experimental data. Second, data collection is fraught with experimental error, and this error must be minimized to the greatest extent possible. Consider an absolute error (see Glossary of Terms) of 1mm. A 1mm absolute error isn’t much when one is considering the distance, say, between the floor and the ceiling in a room.

The amount of relative error (see Glossary of Terms) is small. A 1mm absolute error is huge, for instance, if you are trying the measure the size of a tiny ball bearing. To increase the accuracy of an experiment it is important to reduce the amount of relative error in the data to a minimum. Step 7. Collect and interpret data. In introductory physics labs this step will generally take the form of graphical analysis (see Relationships from Graphs). Using the computer program Graphical Analysis, graph the independent variable against the dependent variable. Perform regression analysis (after linearizing the data if appropriate) and determine the form of the relationship. Pay careful attention to how well your fit matches your theoretical model. If your theory base is correct, then the only reason there should be a difference between the calculated model and the theoretical model is due to random error in the experiment. When you conduct your regression analysis, keep in mind that a 5th order polynomial will accurately fit almost any data. Fortunately, nature is much simpler that that. So, for instance, look for a trigonometric fit rather than a 5th order polynomial fit if you identify an unusual curve. Additionally, be certain that you create a physically reasonable regression model. That is, if the variable on the x-axis is zero and you expect that the variable on the y-axis must similarly be zero, then the regression line must past through the origin (e.g., conduct regression analysis using a proportion fit, y = mx, instead of a linear fit, y = mx + b). Convert the algebraic expression into one that is physically interpreted. Replace y by the variable plotted on the y-axis, and x by the variable plotted on the x-axis. Determine the value and units of the slope and the y-intercept. Give these a physical interpretation if possible. Step 8. Consider the overall precision of the experiment. If your experimental results in no way match your theoretical model (if used), then there are two sources of possible error. Either your model is wrong, or you data collection procedures contain a systematic error (see Glossary of Terms). You might be asked to use methods dealing with the propagation of error to determine the amount of expected error in a calculated term on the basis of the errors in the experimental terms. Making Measurements

Errors, Accuracy, and Precision

Inmakingphysicalmeasurements,oneneedstokeepinmindthatmeasurementsarenotcompletelyaccurate.Eachmeasurementwillhavesomenumberofsignificantfiguresandshouldalsohavesomeindicationastohowmuchwecan“trust”it(i.e.errorbars).Thusinordertoreliablyinterpretexperimentaldata,weneedtohavesomeideaastothenatureofthe“errors”associatedwiththemeasurements.Therearewholetextbooksdevotedtoerroranalysis.Wewillnotevenattempttobecompleteinourdiscussionofthetopic,ratherwewillpresentsomebasicideastoassistyouinanalyzingyourdata.Whendealingwitherroranalysis,itisagoodideatoknowwhatwereallymeanbyerror.Tobeginwith,let’stalkaboutwhaterrorisnot.Errorisnotablundersuchasforgettingtoputthedecimalpointinrightplace,usingthewrongunits,transposingnumbers,oranyothersillymistake.Errorisnotyourlabpartnerbreakingyourequipment.Errorisn’teventhedifferencebetweenyourownmeasurementandsomegenerallyacceptedvalue.(Thatisadiscrepancy.)Acceptedvaluesalsohaveerrorsassociatedwiththem;theyarejustbettermeasurementsthanyouwillbeabletomakeinathree‐hourundergraduatephysicslab.Whatwereallymeanbyerrorhastodowithuncertaintyinmeasurements.Noteveryoneinlabwillcomeupwiththesamemeasurementsthatyoudoandyet(withsomeobviousexceptionsduetoblunders)wemaynotgivepreferencetooneperson’sresultsoveranother’s.Thusweneedtoclassifytypesoferrors.Generallyspeaking,therearetwotypesoferrors:1)systematicerrorsand2)randomerrors.Systematicerrorsareerrorsthatareconstantandalwaysofthesamesignandthusmaynotbereducedbyaveragingoveralotofdata.Examplesofsystematicerrorswouldbetimemeasurementsbyaclockthatrunstoofastorslow,distancemeasurementsbyaninaccuratelymarkedmeterstick,currentmeasurementsbyinaccuratelycalibratedammeters,etc.Generallyspeaking,systematicerrorsarehardtoidentifywithasingleexperiment.Incaseswhereitisimportant,systematicerrorsmaybeisolatedbyperformingexperimentsusingdifferentproceduresandcomparingresults.Iftheproceduresaretrulydifferent,the

systematicerrorsshouldalsobedifferentandhopefullyeasilyidentified.Anexperimentthathasverysmallsystematicerrorsissaidtohaveahighdegreeofaccuracy.Randomerrorsareawholedifferentbag.Theseerrorsareproducedbyanyoneofanumberofunpredictableandunknownvariationsintheexperiment.Examplesmightincludefluctuationsinroomtemperature,fluctuationsinlinevoltage,mechanicalvibrations,cosmicrays,etc.Experimentswithverysmallrandomerrorsaresaidtohaveahighdegreeofprecision.Sincerandomerrorsproducevariationsbothaboveandbelowsomeaveragevalue,wemaygenerallyquantifytheirsignificanceusingstatisticaltechniques. Dealing with Uncertainties

Uncertainty in Measurement When scientists collect data, they rarely make just one measurement and leave it at that. Random error and unanticipated events can cause a single data point to be somewhat non-representative. Thus, multiple measurements are usually taken, and the “best value” is represented using a variety of approaches. Mean, median, and mode are measurement of central tendency. The degree to which data tend to be distributed around the mean value is known as variation or dispersion. A variety of measures of dispersion or variation around a mean are available, each with its own peculiar idiosyncrasy. The most common measures are the following: range, mean deviation, standard deviation, semi-interquartile range, and 10-90 percentile range. In this brief article we will examine only those measures most suitable for the purposes of an introductory lab experience. As a scientist or an engineer you need to know how to deal effectively with uncertainty in measurement. Uncertainty in measurement is not to be confused with significant digits that have their own rules of uncertainty (see Significant Figures in Measurement and Computation elsewhere in this Handbook). When one reads that some measurement has the value and absolute error of, say, d= 3.215m± .003m, how is this to be interpreted? The value 3.215m is known as the arithmetic mean (as opposed to weighted mean, geometric mean, harmonic mean, etc.) It is sometimes loosely called the average (but there are different types of averages as well). The arithmetic mean of N measurements is defined as follows:

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x =1N

xii=1

N

∑

Unless the person who provides the information indicates what the absolute error represents, there is usually no way of knowing. Does the indicated error represent the range of the values measured? That is, does this merely indicate that the “true” value of d is somewhere between 3.212m and 3.218m? It could imply this, but other interpretations are also possible. Inanothercase,±0.003mcouldrepresentthemeandeviation.Themeandeviationisthemeanoftheabsolutedeviationsofasetofdataaboutthedata’smean.Themeandeviationisdefinedmathematicallyas

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MD =1N

xi − xi=1

N

∑

where

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x is the arithmetic mean of the distribution. In yet another case ±0.003m could represent one standard deviation. This would then represent a probability that there is a 68% chance (±1σ or ±1standard deviation) that the “best” value is within the range of 3.212m and 3.218m. Such an interpretation is not strictly justified unless more than 30 data points have been used to calculate the mean value and standard deviation. In this interpretation data must also be “normally distributed” around the mean. That is, the mean, median, and mode of the data must be essentially the same. There must be a “bell shaped” distribution of data that are symmetrically distributed around the mean. Most data collected in introductory labs fail to meet these criteria. Whatever form of error representation you use in lab, you must be consistent in your expressions of uncertainty in

measurement. You must also declare what your error represents. For the purposes of these introductory labs, the following forms of uncertainty in measurement will be used if called for. Example 1: Range – You use a meter stick with divisions marked off in 1mm units. The measurements of length appear to be somewhere between 93mm and 94mm. You would best represent the value of the length with uncertainty as follows stating without any ambiguity that the true value of the length measured lay somewhere between 93mm and 94mm.

93.5mm ± 0.5mm Example 2: Mean Deviation – You make six observations using an electronic photogate timer to measure nearly identical falls of a ball with the following results:

Observation 1: 6.556s Observation 2: 6.576s Observation 3: 6.630s Observation 4: 6.587s Observation 5: 6.575s Observation 6: 6.602s

The value of the mean deviation is calculated thus:

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x= 6.588s |x1-

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x |=0.022s |x2-

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x |=0.012s |x3-

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x |=0.042s |x4-

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x |=0.001s |x5-

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x |=0.013s |x6-

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x |=0.014s Sum of deviations = 0.104s

Mean deviation = 0.017s

The mean of the data along with mean deviation are accurately stated as 6.588s ± 0.017s Example 3: Standard Deviation – Let’s say you have made 30 measurements using your electronic timer. For the purposes of these introductory labs we will use the mean ±1standard deviation (or ±1σ) to represent the error. This process makes three major assumptions – that the variation in the data comes only from random error, that 30 or so data points are sufficient to calculate a good standard deviation, and that data constitute a “normal” (bell-shaped distribution) around the mean. None of these assumptions might be correct in a particular situation, so it might be a good idea to examine the distribution of data to make certain that data have the standard symmetrical “bell shaped” curve. Thus, interpretation of the standard deviation must be done with great caution. Standard deviation of N data points can be readily calculated with the use of the following formula:

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σ =1N

(xi − x)2

i=1

N

∑

Most scientific calculators will, given the data points, readily calculate the standard deviation with ease. Lab students should use this form to express uncertainty in repeated measures throughout these introductory lab activities when called for. Due caution should be observed in the use of significant digits.

Absolute and Relative Error Absolute and relative error are two types of error with which every experimental scientist should be familiar. The differences are important.

Absolute Error: Absolute error is the amount of physical error in a measurement, period. Let’s say a meter stick is used to measure a given distance. The error is rather hastily made, but it is good to

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±1mm . This is the absolute error of the measurement. That is,

absolute error =

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±1mm (0.001m).

In terms common to Error Propagation

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absolute error = Δx where x is any variable. Relative Error: Relative error gives an indication of how good a measurement is relative to the size of the thing being measured. Let’s say that two students measure two objects with a meter stick. One student measures the height of a room and gets a value of 3.215 meters

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±1mm (0.001m). Another student measures the height of a small cylinder and measures 0.075 meters

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±1mm (0.001m). Clearly, the overall accuracy of the ceiling height is much better than that of the 7.5 cm cylinder. The comparative accuracy of these measurements can be determined by looking at their relative errors.

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relative error =absolute error

value of thing measured

or in terms common to Error Propagation

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relative error =Δxx

where x is any variable. Now, in our example,

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relative errorceiling height =0.001m3.125m

•100 = 0.0003%

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relative errorcylinder height =0.001m0.075m

•100 = 0.01%

Clearly, the relative error in the ceiling height is considerably smaller than the relative error in the cylinder height even though the amount of absolute error is the same in each case.

Propagation of Errors Let’s say that you have experimentally measured two quantities that will be used to determine a third quantity. There are errors in each of the measured quantities. How does the error in each of these two measured quantities affect the value of the calculated quantity? Assume for the sake of the discussion that we use a pendulum to indirectly measure the acceleration due to gravity according to the hypothetical relationship

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t = 2π lg

The length of the pendulum, l, and the period of oscillation, t, are measured. In the process of determining the local value of g, there is an error in the measurement of the length of the pendulum Δl. Similarly, assume an error in the

measured period equal to Δt. How do these errors in length and period affect the resulting value of g? Consider the following process of analyzing error propagation. Given the above relationship that has reformulated to eliminate the square root and terms in the denominator, start by substituting g + Δg for g, t + Δt for t, and l + Δl for l.

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gt 2 = 4π 2l

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(g+Δg)(t +Δt)2 = 4π 2(l +Δl)(g+Δg)(t2 +2tΔt +Δt2) = 4π 2(l +Δl)

gt2 +2gtΔt + gΔt2 +Δgt2 +2tΔgΔt +ΔgΔt2 = 4π 2l + 4π 2Δl

Eliminating terms in double and triple Δ (assumed << 1) and canceling like terms (note also that

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gt 2 = 4π 2l) results in

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Δg =4π 2Δl − 2gtΔt

t 2

Now, because the Δ terms can be considered either +/-, the maximum possible error in g will result when both error terms in the numerator are of like sign. That is, the maximum absolute error is given by the relationship

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Δgmax =4π 2Δl + 2gtΔt

t 2

After rearranging our terms and making a substitution based on our original identity (

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gt 2 = 4π 2l) we get the relative error

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Δgg

max

=Δll

+2Δtt

Additional pointers for making this process work well:

1) Most of the time it is best to simplify a relationship before making the error term substitutions. Eliminating square roots and terms in the denominator will make the task of finding the error term much simpler. For instance, consider the following range equation with error terms in s, v, h, and g. Convert the equation before beginning the required substitutions:

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s = v × 2hg

becomes

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s2g = 2v 2h then

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(s+Δs)2(g+Δg) = 2(v +Δv)2(h +Δh)

2) Pay close attention to substitutions that can be made by manipulating the simplified original equation obtained from following the previous hint (e.g.

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2s = gt2). The reason for these substitutions is to eliminate “like” terms from both sides of the equation. This allows one to find terms that commonly appear in error propagation equations. In the example immediately above, this would mean the following:

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Δvv

,

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Δss

,

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Δgg

, and

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Δhh

It should be noted that with knowledge of partial differential equations, equations for absolute and relative error can be much more easily derived (especially with complex equations).

Percent Difference – Percent Error Sometimes scientists will want to compare their results with those of others, or with a theoretically derived prediction. Each of these types of comparisons call for a different type of analysis, percent difference and percent error respectively. Percent Difference: Applied when comparing two experimental quantities, E1 and E2, neither of which can be considered the “correct” value. The percent difference is the absolute value of the difference over the mean times 100.

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% Difference =E1 − E2

12E1 + E2( )

•100

Percent Error: Applied when comparing an experimental quantity, E, with a theoretical quantity, T, which is considered the “correct” value. The percent error is the absolute value of the difference divided by the “correct” value times 100.

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% Error =T − ET

•100

Significant Figures

Inmakingphysicalmeasurements,oneneedstokeepinmindthatmeasurementsarenotcompletelyaccurate.Eachmeasurementwillhavesomenumberofsignificantfiguresandshouldalsohavesomeindicationastohowmuchwecan“trust”it(i.e.errorbars).Thusinordertoreliablyinterpretexperimentaldata,weneedtohavesomeideaastothenatureofthe“errors”associatedwiththemeasurements.Therearewholetextbooksdevotedtoerroranalysis.Wewillnotevenattempttobecompleteinourdiscussionofthetopic,ratherwewillpresentsomebasicideastoassistyouinanalyzingyourdata.SignificantFiguresThesignificantfiguresinaquantityarethemeaningfuldigitsinit.Thereareseveralconventionslistedbelowtokeepinmindwhendealingwithsignificantfigures:1) Anynonzerodigitisasignificantdigit.Examples:3.14hasthreesignificantdigitsand239.67hasfive

significantdigits.2) Zerosinbetweennonzerovaluesaresignificantdigits.Example20.4hasthreesignificantdigits.3) ZerostotheleftofthefirstnonzerodigitareNOTsignificant.Example:0.0012hastwosignificantdigits.4) Fornumberswithdecimalpoints,zerostotherightofnonzerodigitsaresignificant.Example:Both2.0

and0.0020havetwosignificantdigits.5) Fornumberswithoutdecimalpoints,zerostotherightofnonzerodigitsmayormaynotbesignificant.

Example:350mayhavetwoorthreesignificantdigits.Toavoidconfusion,useadecimalpoint:350.hasthreesignificantdigits,350hastwo.

6) Thelastsignificantdigitinanumbershouldhavethesameorderofmagnitudeastheuncertainty.Examples:3.1415±0.0001;0.8±0.2

7) Theuncertaintyshouldberoundedtooneortwosignificantdigits.8) When“doing”mathematicalmanipulationswithnumbersthathavevaryingnumbersofsignificant

digits,thefinalresultcannothavemoresignificantdigitsthantheleastwellknownnumber.Inpractice,wewillkeepallthenumberstotheirknownaccuracythroughoutthecalculationandthenroundtothenumberofsignificantfiguresattheend.Examples:2.0*3.14159=6.3;25.001+32.0=55.0

Conversion Factors

The use of conversion factors will not be confusing so long as certain rules are adhered to rigidly. These rules can best be shown with the use of examples: Problem: Convert 3 (assumed an integer) inches (in) to millimeters (mm). The conversion factor is 1in = 25.4mm precisely. This can be rewritten in two ways:

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1in25.4mm

=1 or

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25.4mm1in

=1

Now, we can multiply any quantity by 1 without changing its essential value. We can multiply 3 inches by 1 and still have 3 inches. If we choose to rewrite 1 in one of the above forms, we can convert a distance in one system of measure (British) to that of another (metric). The numerical value will not be the same because the unit of measurement will change. However, the actual distance will be the same in either system of measurement. The question is, which one of the above two ways should we choose to write 1 so as to successfully convert from one system of measurement to another? If we choose to write 1 in the first way, we have:

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3in = 3in ×1= 3in × 1in25.4mm

=0.118in2

mm

The result is in mixed units that have no direct physical meaning and is a combination of both metric and British systems. If we choose to write 1 in the second way, we have:

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3in = 3in ×1= 3in × 25.4mm1in

= 76.2mm

Note that the inches unit has canceled; the result is purely metric. What happens if one needs to convert, say, cubic centimeters (cm

3) into cubic meters (m

3)? Consider the following

problem. Problem: Convert 300cm

3 into cubic meters. The conversion factor is 100cm = 1 meter. Now, one cannot merely multiply 300cm

3 by 1m/100cm to arrive at the answer. If this were done, the answer would come out in cm

2 x m and, again, mixed units. To handle this sort of complication, we must rewrite 1 as 1

3. One cubed is the same as one. For example:

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300cm3 = 300cm3 ×13 = 300cm3 × ( 1m100cm

)3

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= 300cm3 ×1m3

1×106cm3 =300m3

106= 3×10−4m3

When using conversion factors, always pay meticulous attention to the units and their associated powers. In doing so, you need never make an error in converting from one system to another.

Dimensional Analysis Scientists sometimes will use a process called dimensional analysis to predict the form of the relationship between variables in a system. While this does NOT constitute a theory base, it does give scientists a better understanding of what to expect experimentally, and helps them to design scientific experiments. Carefully observe the following process of dimensional analysis for a cart accelerating down a fixed inclined plane. Let’s say that a scientist observes the motion of the cart on the fixed incline, and after conducting a few qualitative experiments concludes that the distance of the cart is a function of its acceleration and the time. Assume that the cart accelerates from rest when the stopwatch starts. What is the expected form of the relationship between distance, acceleration, and time? The form of this relationship can be predicted using an approach known as dimensional analysis. This process is based on the knowledge that the distance (d) is a function of the acceleration (a) and the time (t). That is,

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d = f (a,t)

Now, if d (expressed in meters, m) is related to both a (expressed in meters per second squared, m/s2) and t (expressed in seconds, s) in some form, then the only thing missing is a proportionality constant and the powers of the variable terms. Write a proportionality applying power terms x and y to a and t respectively. Replace variables by units and solve for the power terms as, in this case, two simultaneous equations with two unknowns. Working backward, then find the form of the equation and insert proportionality constant. (Note that if the unit of time, s, is present on the right side of the equation, it must also be present on the left side of the equation. Place s0 on the left side of the equation in step two below as shown; s0 is equal to 1 and does not affect the equality.)

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d ∝ axty

replacing variables with units

m1s0 ∝ms2

x

sy

simplifyingm1s0 ∝mxsy−2x

equating exponents on m and s1 = x0 = y −2xand solving simultaneous equationsy = 2xhence, y = 2thus, d ∝ a1t2

or, d = kat2

Experiments can be conducted or theoretical work performed to find the value of the constant, k. In reality, k equals ½. This gives the familiar kinematic equation

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d =12at2

Practices in the Laboratory Evaluating an Experiment

Chi-Square Test for Goodness of Fit (after Applied Statistics by Hinkle/Wiersma/Jurs)

Scientists will often use the Chi-square (

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χ 2) test to determine the goodness of fit between theoretical and experimental data. In this test, we compare observed values with theoretical or expected values. Observed values are those that the researcher obtains empirically through direct observation; theoretical or expected values are developed on the basis of some hypothesis. For example, in 200 flips of a coin, one would expect 100 heads and 100 tails. But what if 92 heads and 108 tails are observed? Would we reject the hypothesis that the coin is fair? Or would we attribute the difference between observed and expected frequencies to random fluctuation?

Consider another example. Suppose we hypothesize that we have an unbiased six-sided die. To test this hypothesis, we roll the die 300 times and observe the frequency of occurrence of each of the faces. Because we hypothesized that the die is unbiased, we expect that the number on each face will occur 50 times. However, suppose we observe frequencies of occurrence as follows:

Face value Occurrence

1 42 2 55 3 38 4 57 5 64 6 44

Again, what would we conclude? Is the die biased, or do we attribute the difference to random fluctuation? Consider a third example. The president of a major university hypothesizes that at least 90 percent of the teaching and research faculty will favor a new university policy on consulting with private and public agencies within the state. Thus, for a random sample of 200 faculty members, the president would expect 0.90 x 200 = 180 to favor the new policy and 0.10 x 200 = 20 to oppose it. Suppose, however, for this sample, 168 faculty members favor the new policy and 32 oppose it. Is the difference between observed and expected frequencies sufficient to reject the president's hypothesis that 90 percent would favor the policy? Or would the differences be attributed to chance fluctuation?

Lastly,consideranexperimentalresultwhere,forgivenindependentvaluesof“X,”thefollowingtheoretical(expected)andexperimental(observed)dependentvaluesofYwerefound:

X Ytheoretical Yexperimental 3.45 11.90 11.37 4.12 16.97 17.02 4.73 22.37 23.78

5.23 27.35 26.13 6.01 36.12 35.96 6.82 46.51 45.22 7.26 52.71 53.10

In each of these examples, the test statistic for comparing observed and expected frequencies is

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χ 2, defined as follows:

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χ 2 =(O− E)2

Ei=1

k∑

where

O = observed value E = expected value

k = number of categories, groupings, or possible outcomes The calculations of

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χ 2for each of the three examples, using the above formula, are found in Tables 1-4. TABLE 1. Calculation of

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χ 2 for the Coin-Toss Example

Face O E O-E (O-E)2 (O-E)2/E Heads 92 100 -8 64 .64 Tails 108 100 +8 64 .64

Totals 200 200 0 -- 1.28 =

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χ 2 TABLE 2. Calculation of

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χ 2 for the Die Example

Face Value O E O-E (O-E)2 (O-E)2/E 1 42 50 -8 64 1.28 2 55 50 5 25 .50 3 38 50 -12 144 2.88 4 57 50 7 49 .98 5 64 50 14 196 3.92 6 44 50 -6 36 .72

Totals 300 300 0 -- 10.28=

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χ 2 TABLE 3. Calculation of

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χ 2 for the Consulting-Policy Example

Disposition O E O-E (O-E)2 (O-E)2/E Favor 168 180 -12 144 .80

Oppose 32 20 +12 144 7.20 Totals 200 200 0 -- 8.00=

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χ 2

TABLE 4. Calculation of

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χ 2 for the Experiment Example

X O E O-E (O-E)2 (O-E)2/E 3.45 11.37 11.90 -.53 .28 .02 4.12 17.02 16.97 .05 .02 .00 4.73 23.78 22.37 1.41 1.99 .09 5.23 26.13 27.35 -1.22 1.49 .05 6.01 35.96 36.12 -.16 .03 .00 6.82 45.22 46.51 -1.29 1.66 .04 7.26 53.10 52.71 .39 .15 .00

Totals 212.58 213.93 -1.35 5.62 0.20 =

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χ 2

There is a family of

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χ 2 distributions, each a function of the degrees of freedom associated with the number of categories in the sample data. Only a single degree of freedom (df) value is required to identify the specific

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χ 2distribution. Notice that all values of

€

χ 2are positive, ranging from zero to infinity. Consider the degrees of freedom for each of the above examples. In the coin example, note that the

expected frequencies in each of the two categories (heads or tails) are not independent. To obtain the expected frequency of tails (100), we need only to subtract the expected frequency of heads (100) from the total frequency (200), or 200 - 100 = 100. Similarly, for the example of the new consulting policy, the expected number of faculty members who oppose it (20) can be found by subtracting the expected number who support it (180) from the total number in the sample (200), or 200 - 180 = 20. Thus, given the expected frequency in one of the categories, the expected frequency in the other is readily determined. In other words, only the expected frequency in one of the two categories is free to vary; that is, there is only 1 degree of freedom associated with these examples.

For the die example, there are six possible categories of outcomes: the occurrence of the six faces. Under the assumption that the die is fair, we would expect that the frequency of occurrence of each of the six faces of the die would be 50. Note again that the expected frequencies in each of these categories are not independent. Once the expected frequency for five of the categories is known, the expected frequency of the sixth category is uniquely determined, since the total frequency equals 300. Thus, only the expected frequencies in five of the six categories are free to vary; there are only 5 degrees of freedom associated with this example.

In the last example dealing with the data from the experiment, the cross-tabulation table has 2 columns and 7 rows. The degrees of freedom for such tables is (# rows – 1)(# columns – 1) = (6)(1) = 6 The Critical Values for the

€

χ 2 Distribution The use of the

€

χ 2 distribution in hypothesis testing is analogous to the use of the t and F distributions. A null hypothesis is stated, a test statistic is computed, the observed value of the test statistic is compared to the critical value, and a decision is made whether or not to reject the null hypothesis. For the coin example, the null hypothesis is that the frequency of heads is equal to the frequency of tails. For the die example, the null hypothesis is that the frequency of occurrence of each of the six faces is the same. In general, it is not a requirement for the categories to have equal expected frequencies. For instance, in the example of the new consulting policy, the null hypothesis is that 90 percent of the faculty will support the new policy and 10 percent will not.

The critical values of

€

χ 2 for 1 through 30 degrees of freedom are found in Table 5. Three different percentile points in each distribution are given – α = .10, α = .05, and α = .01. (e.g., chances of 10%, 5%, and 1% respectively of rejecting the null hypothesis when it should be retained). For the coin and consulting-policy examples, the critical values of

€

χ 2 for 1 degree of freedom, with α = .05 and α = .01, are 3.841 and 6.635, respectively. For the die example, the corresponding critical values of

€

χ 2 for 5 degrees of freedom are 11.070 and 15.086. Although Table 5 is sufficient for many research settings in the sciences, there are some situations in which the degrees of freedom associated with a

€

χ 2 test are greater than 30. These situations are not addressed here.

Now that we have seen the table of critical values for the

€

χ 2 distribution, we can complete the examples. For the coin example, the null hypothesis is that the frequency of heads equals the frequency of tails. As we mentioned, because there are only two categories, once the expected value of the first category is determined, the second is uniquely determined. Thus, there is only 1 degree of freedom associated with this example. Assuming that the α = .05 level of significance is used in testing this null hypothesis, the critical value of

€

χ 2 (

€

χCV2 ) is 3.841 (see Table 5). Notice that, in Table 1, the calculated

value of

€

χ 2 is 1.28. Because the calculated value does not exceed the critical value, the null hypothesis (the coin is fair) is not rejected; the differences between observed and expected frequencies are

attributable to chance fluctuation. That is, when the calculated value of

€

χ 2 exceeds the critical value, the data support the belief that a significant difference exists between expected and actual values.

For the example of the new consulting policy, the null hypothesis is that 90 percent of the faculty would support it and 10 percent would not. Again, because there are only two categories, there is 1 degree of freedom associated with the test of this hypothesis. Thus, assuming α = .05, the

€

χCV2 is 3.841. From

Table 3, we see that the calculated value of

€

χ 2 is 8.00; therefore, the null hypothesis (that there is no difference) is rejected. The conclusion is that the percentage of faculty supporting the new consulting policy is not 90.

For the die example, the null hypothesis is that the frequency of occurrence of each of the six faces is the same. With six categories, there are 5 degrees of freedom associated with the test of this hypothesis; the

€

χCV2 for α = .05 is 11.070. Using the data from Table 2,

€

χ 2 = 10.28. Because this calculated value is less than the critical value, the null hypothesis is retained. The conclusion is that the differences between the observed and expected frequencies in each of the six categories are attributable to chance fluctuation.

For the example with the experiment, the null hypothesis is that there is no difference between theoretical and experimental results. With seven data pairs there are 6 degrees of freedom associated with the test of this hypothesis; the

€

χCV2 for α = .05 is 12.592. Because this calculated value of

€

χ 2 (0.20) is less than the critical value (12.592), the null hypothesis is retained. The conclusion is that the differences between the observed and expected values in each of the seven data pairs are attributable to chance fluctuation. The experimental results are therefore consistent with the theoretical results to with a 95% chance of probability.

TABLE 5. CRITICAL

€

χ 2 VALUES FOR UP TO 30 DEGREES OF FREEDOM

df

1 2 3 4 5

6 7 8 9

10

11 12 13 14 15

16 17 18 19 20

21 22 23 24 25

26 27 28 29 30

α = .10 2.706 4.605 6.251 7.779 9.236 10.645 12.017 13.362 14.684 15.987 17.275 18.549 19.812 21.064 22.307 23.542 24.769 25.989 27.204 28.412 29.615 30.813 32.007 33.196 34.382 35.563 36.741 37.916 39.087 40.256

α = .05 3.841 5.991 7.815 9.488 11.070 12.592 14.067 15.507 16.919 18.307 19.675 21.026 22.362 23.685 24.996 26.296 27.587 28.869 30.144 31.410 32.671 33.924 35.172 36.415 37.652 38.885 40.113 41.337 42.557 43.773

α = .01 6.635 9.210 11.345 13.277 15.086 16.812 18.475 20.090 21.666 23.209 24.725 26.217 27.688 29.141 30.578 32.000 33.409 34.805 36.191 37.566 38.932 40.289 41.638 42.980 44.314 45.642 46.963 43.278 49.558 50.892

Graphs

Common Graph Forms in Physics Workingwithgraphs–interpreting,creating,andemploying–isanessentialskillinthesciences,andespeciallyinphysicswhererelationshipsneedtobederived.Asanintroductoryphysicsstudentyoushouldbefamiliarwiththetypicalformsofgraphsthatappearinphysics.BelowareanumberoftypicalphysicalrelationshipsexhibitedgraphicallyusingstandardX‐Ycoordinates(e.g.,nologarithmic,power,trigonometric,orinverseplots,etc.).Studytheformsofthegraphscarefully,andbepreparedtousetheprogramGraphicalAnalysistoformulaterelationshipsbetweenvariablesbyusingappropriatecurve‐fittingstrategies.Notethatallnon‐linearformsofgraphscanbemadetoappearlinearby“linearizing”thedata.LinearizationconsistsofsuchthingsasplottingXversusY2orXversus1/YorYversuslog(X),etc.Note:Whilea5thorderpolynomialmightgiveyouabetterfittothedata,itmightnotrepresentthesimplestmodel.

X0.800 1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00

Y

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

10.0

11.0

12.0

13.0

LINEARRELATIONSHIP:Whathappensifyougetagraphofdatathatlookslikethis?HowdoesonerelatetheXvariabletotheYvariable?It’ssimple,Y=A+BXwhereBistheslopeofthelineandAistheY‐intercept.ThisischaracteristicofNewton’ssecondlawofmotionandofCharles’law:

F = ma

PT= const.

X0.00 0.500 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

0.00

0.400

0.800

1.20

1.60

2.00

2.40

2.80

3.20

3.60

4.00

4.40

4.80

5.20

Y

INVERSERELATIONSHIP:Thismightbeagraphofthepressureandtemperatureforachangingvolumeconstanttemperaturegas.Howwouldyoufindthisrelationshipshortofusingacomputerpackage?Theansweristosimplifytheplotbymanipulatingthedata.PlottheYvariableversustheinverseoftheXvariable.Thegraphbecomesastraightline.TheresultingformulawillbeY=A/XorXY=A.ThisistypicalofBoyle’slaw:

PV = const.

X0.400 0.800 1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00

Y

0.00

0.400

0.800

1.20

1.60

2.00

2.40

2.80

3.20

3.60

4.00

INVERSE‐SQUARERELATIONSHIP:OftheformY=A/X2.CharacteristicofNewton’slawofuniversalgravitation,andtheelectrostaticforcelaw:

F =Gm1m2

r2

F =kq1q2r 2

Inthelattertwoexamplesabovethereareonlysubtledifferencesinform.Manycommongraphformsinphysicsappearquitesimilar.Onlybelookingatthe“RMSE”(rootmeansquareerror

providedinGraphicalAnalysis)canoneconcludewhetheronefitisbetterthananother.ThebetterfitistheonewiththesmallerRMSE.Seepage2formoreexamplesofcommongraphformsinphysics.

DOUBLE‐INVERSERELATIONSHIP:Oftheform1/Y=1/X+1/A.Mostreadilyidentifiedbythepresenceofanasymptoticboundary(y=A)withinthegraph.Thisformischaracteristicofthethinlensandparallelresistanceformulas.

€

1f

=1di

+1doand 1

Rt=1R1+1R2

X0.00 0.200 0.400 0.600 0.800 1.00 1.20 1.40 1.60 1.80 2.00

Y

0.00

0.400

0.800

1.20

1.60

2.00

2.40

2.80

3.20

3.60

4.00

POWERRELATIONSHIP:OftheformY=AXB.Typicalofthedistance‐timerelationship:

d =12at2

X0.00 0.400 0.800 1.20 1.60 2.00 2.40 2.80 3.20

Y

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

10.0

11.0

12.0

13.0

POLYNOMIALOFTHESECONDDEGREE:OftheformY=AX+BX2.Typicalofthekinematicsequation:

d = vot +12at2

X0.200 0.400 0.600 0.800 1.00 1.20 1.40 1.60 1.80 2.00

Y

1.20

1.60

2.00

2.40

2.80

3.20

3.60

4.00

4.40

4.80

5.20

5.60

6.00

6.40

6.80

7.20

7.60

EXPONENTIALRELATIONSHIP:OftheformY=A*exp(BX).Characteristicofexponentialgrowthordecay.Graphtoleftisexponentialgrowth.Thegraphofexponentialdecaywouldlooknotunlikethatoftheinverserelationship.Characteristicofradioactivedecay.

€

N = Noe−λt

X0.500 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

Y

-0.800

-0.600

-0.400

-0.200

0.00

0.200

0.400

0.600

0.800

1.00

1.20

1.40

1.60

1.80

NATURALLOG(LN)RELATIONSHIP:OftheformY=Aln(BX).Characteristicofentropychangeduringafreeexpansion:

Sf − Si = nRlnV f

Vi

GraphinginPhysics(Version2)

CommentsbyCarlJ.WenningWorkingwithgraphs‐‐interpreting,creating,andemploying‐‐isanessentialskillinthesciences,andespeciallyinphysicswhererelationshipsneedtobederived.Asaphysicsstudentyouprobablyhaveusedgraphinghundredsoftimesandareprettymuchfamiliarwithboththeinterpretationandcreationofgraphs.Nonetheless,myexperienceinPHY302isthatmoststudentsareunawareofhowpowerfulistheusegraphstodeterminerelationshipsbetweenvariables.Belowareanumberoftypicalphysicalrelationshipsexhibitedgraphically.Studytheformsofthegraphscarefullyandbepreparedtohelpyourstudentsformulaterelationshipsbetweenvariables.

X0.800 1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00

Y

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

10.0

11.0

12.0

13.0

LINEARRELATIONSHIP:Whathappensifyougetagraphofdatathatlookslikethis?HowdoesonerelatetheXvariabletotheYvariable?It’ssimple,Y=A+BXwhereBistheslopeofthelineandAistheY‐intercept.CharacteristicofNewton’ssecondlawandofCharles’law:

F = ma

PT= const.

X0.00 0.500 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

0.00

0.400

0.800

1.20

1.60

2.00

2.40

2.80

3.20

3.60

4.00

4.40

4.80

5.20

Y

INVERSERELATIONSHIP:Thismightbeagraphofthepressureandtemperatureforachangingvolumeisothermalgas.Howwouldyoufindthisrelationshipshortofusingacomputerpackage?Theansweristolinearizethedata.PlottheYvariableversus1overtheXvariable.Thegraphbecomesastraightline.TheresultingformulawillbeY=A/XorXY=A.ThisistypicalofBoyle’slaw:

PV = const.

X0.400 0.800 1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00

Y

0.00

0.400

0.800

1.20

1.60

2.00

2.40

2.80

3.20

3.60

4.00

INVERSE‐SQUARERELATIONSHIP:OftheformY=A/X2.CharacteristicofNewton’slawofuniversalgravitationoftheelectrostaticlaw:

F =Gm1m2

r2

F =kq1q2r 2

DOUBLE‐INVERSERELATIONSHIP:Oftheform1/Y=1/X+1/A.Mostreadilyidentifiedbythepresenceofanasymptoticboundary(y=A)withinthegraph.Characteristicofthinlensandparallelresistanceformulas.

€

1f

=1di

+1doand 1

Rt=1R1+1R2

X0.00 0.200 0.400 0.600 0.800 1.00 1.20 1.40 1.60 1.80 2.00

Y

0.00

0.400

0.800

1.20

1.60

2.00

2.40

2.80

3.20

3.60

4.00

POWERRELATIONSHIP:OftheformY=AXB.Typicalofthedistance‐timerelationship:

d =12at2

X0.00 0.400 0.800 1.20 1.60 2.00 2.40 2.80 3.20

Y

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

10.0

11.0

12.0

13.0

POLYNOMIALOFTHESECONDDEGREE:OftheformY=AX+BX2.Typicalofthekinematicsequation:

d = vot +12at2

X0.200 0.400 0.600 0.800 1.00 1.20 1.40 1.60 1.80 2.00

Y

1.20

1.60

2.00

2.40

2.80

3.20

3.60

4.00

4.40

4.80

5.20

5.60

6.00

6.40

6.80

7.20

7.60

EXPONENTIALRELATIONSHIP:OftheformY=A*exp(BX).Characteristicofexponentialgrowthordecay.Graphtoleftisexponentialgrowth.Thegraphofexponentialdecaywouldlooknotunlikethatoftheinverserelationship.Characteristicofradioactivedecay.

€

N = Noe−λt

X0.500 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

Y

-0.800

-0.600

-0.400

-0.200

0.00

0.200

0.400

0.600

0.800

1.00

1.20

1.40

1.60

1.80

NATURALLOG(LN)RELATIONSHIP:OftheformY=Aln(BX).Characteristicofentropychangeduringafreeexpansion:

Sf − Si = nRlnV f

Vi

EXAMPLES OF COMMON GRAPH CONSTRUCTION ERRORS Datacollectionerrors.Ifyouwantstraight‐linedata,collectonlydataforonlytwopoints;ifreproducibilityisaproblem,conductonlyonetest.Astraightlinecanalwaysbemadetofitthroughtwopoints.Clearlythisisawrongwaytocollectandinterpretdatafor,indeed,alineand,indeed,anycurvecanbemadetopassthroughjusttwopoints.Observationsshouldberepeatedforthesakeofaccuracy.Confusingindependentanddependentvariables.TheIndependentvariableiswhatIchange.”ThesizeoftheDependentvariableDependsonthevalueoftheindependentvariable.Puttingwrongvariablesonaxes.Foramatterofconvenience,itiscommonpracticetoputtheindependentvariableonthehorizontalx‐axis(bottom)ratherthantheverticaly‐axis(side)whenseekingarelationshiptodefinethedependentvariable.Forinstance,ifonewantstoarriveatarelationshipdescribingFasafunctionofa,thenF(thedependentvariable)shouldbeplottedonthey‐axis.Theslopebecomestheproportionalityconstant,F=ma.Failuretounderstandthesignificanceof“linearizing”data.Whendataarenon‐linear(notinastraightlinewhengraphed),itisbestto“linearize”thedata.Thisdoesnotmeantofitthecurveddatapointswithastraightline.Rather,itmeanstomodifyoneofthevariablesinsomemannersuchthatwhenthedataaregraphedusingthisnewdataset,theresultingdatapointswillappeartolieinastraightline.Forinstance,saythedataappeartobeaninversefunction–asxisdoubled,yishalved.Tolinearizethedataforsuchafunctionplotxversus1/y.Ifthisisindeedaninversefunction,thentheplotofxversus1/ydatewillbelinear.Failuretoproperlyrelatey=mx+btothelinearizeddata.Whenplotting,say,distance(onthey‐axis)versustime(onthex‐axis),thecorrectrelationshipbetweendistanceandtimecanbefoundbyrelatingytodistance,xtotime,mtotheslope,andbtothey‐intercept.Forinstance,datahavebeenlinearizedforthefunctionresultinginastraight‐linegraphwhendistanceisplottedversustime‐squared.Theslopeis2m/s2andthey‐interceptis1m.Thecorrectformoftherelationshipbetweenallthevariableswillbedistance=(2m/s2)*time2+1m.Failuretoapplytheappropriatephysicalmodeltothedata.Whenstudentshaveascatteringofdata,theymightatfirstbetemptedtofitthedatawithalinearmodel,y=mx+b.Usingsuchalinearfitwillresultinanerrorifwhenyequals0,xmustalsobe0.Forinstance,iftheforceonaspringis0,itsextensionfromtheequilibriumpositionwillbezero.Thisis,noforce,noextension.SucharelationshipexpressedgraphicallyMUSTpassthroughtheorigin.Becauseoferrorsinthecollecteddata,alinearfitwillalmostneverpassthroughtheoriginasitshould.Thewaytosolvethisproblemistoconductaproportionalfit,y=mx.Thisforcesb=0andassuresthatthebestfitlinewillpassthroughtheorigin.Usingmultiple(0,0)pointstoforcebest­fitlinethroughorigin.Sometimesstudentsjustifiablyrealizethatabest‐fitlineforastraight‐linerelationshipshouldpassthroughtheorigin(e.g.,ifforceequalszero,thenaccelerationmustalsoequalzero).Theythenincludemultiple(0,0)pointsinthedatasetinanefforttoforcethebest‐fitlinethroughtheorigin.Thisrarelyworks.Betteristofitthelinearizeddatawithaproportionalrelationship(y=mx)inthesecasesratherthanalinearrelationship(y=mx+b).Thevalueofb(they‐intercept)isforcedtobezerointheproportionalfit.Failuretoapplyappropriatelabeling.Eachgraphshouldbeappropriatelylabeled;eachaxisshouldbesimilarlylabeledwithitsvariableandunits(inparentheses).Forinstance,time(seconds)ordistance(meters).

Confusingvariableswithunits.Don’tconfusevariables(time,resistance,distance)withtheirunits(seconds,ohms,meters).Connectingdatapointswithstraightlines.Neverconnectevenlinearizeddatapoint‐to‐pointwithstraightlines.“Mothernatureisnotjerky.”Linearizeddatasometimesappearsnottofitthebest‐fitlineprecisely.Thisisfrequentlyduetoerrorsinthedata.Scalingerrors.Inphysicsitisoftenbesttoshowtheorigin(0,0)ineachgraph.Whilethisisnotanerror,itfrequentlyleadstomisinterpretationsofrelationships.Anotherproblemtoavoidisnotstickingwithaconsistentscaleonanaxis.Valuesshouldbeequallyspacedongraphs.Forinstance,thedistancesbetween1,2,3,and4shouldbethesame.Tobeavoidedaresituationswherethespacingbetweensuchnumbersisirregularorinconsistent.Bewarethehighorderpolynomialfit.MotherNaturetendstobequitesimple,butany4thor5thorderpolynomialfunctioncanbemadetofitnearlyanygivensetofdata.Avoidfittingdatawithnon‐linearfunctionsofabovesecondorder(e.g.,y=a+bx+cx2).Simplefunctionslikex=constant/yorx=A*sin(θ)aremuchmorecommon.EXAMPLES OF COMMON GRAPH INTERPRETATION ERRORS Failingtonotethatwhentheslopeiszero(horizontalline),thereisnorelationship.Sometimesstudentsthinkthatarelationshipbetweentwovariablesexistswhen,infact,itdoesnot.Thisoftenhappenswithauto‐scalingcomputergraphingprogramswhere,forinstance,theverticalaxisrunsfrom0.05to0.06andhorizontalaxisreadsfrom0to10.The0.05‐0.06rangeoftheverticalaxisisautomaticallystretchedouttotakeupthesamelinearspaceasthehorizontalaxisthus“fillingthepage.”Inthiscasetheactualrelationshipmightbey=0.0001x.Somefalselythinkthatthisconstitutesameaningfulrelationship.Itisanextremelyweakrelationshipprobablyarisingfrommeasurementerrors.Byrescaleeachaxis,thegraphclearlyshowsalineofessentiallyzeroslope.Failingtoproperlyinterprethorizontalandverticalslopes.Whentheslopeofalineiszerodegrees(horizontal),thenthereisnorelationshipbetweentheplottedvariables.Whentheslopeofalineis90degrees(vertical),thereisnorelationshipbetweentheplottedvariables.Inboththesecases,theso‐called“dependent”variabledoesnotactuallydependupontheindependentvariable.Failingtoproperlyinterpretanegativeslope.AnegativeslopedoesNOTimplyaninverserelationship.Whatitdoesimplyisthatwhenxgetslarger,ybecomeslargerbutinthenegativedirection.y=‐mxisanegativeslope.xy=morx=m/yisaninverserelationship.Additionally,anegativeaccelerationdoesnotnecessaryimplyaslowingdownofanobject.Anobjectmovinginthe–xdirectionwill,infact,beincreasinginspeed.Failingtoproperlyinterpretthey­intercept.Thevalueofthey‐interceptrepresentsthevalueofthedependentvariablewhentheindependentvariableiszero.Sometimeswhenlinesorcurverepresentingrelationshipsclearlyshouldpassthroughtheorigin(0,0),theydonot.Thevalueofthey‐interceptisthensmallbutnotzero.Thisisusuallytheresultofdatacollectionerrors.Linearizeddatacanbe“forcedthroughtheorigin”byconductingaproportionalfitofthedata(y=mx)ratherthanalinearfit(y=mx+b).FindingslopeusingtwodatapointsNOTonbest­fitline.Findingtheslopeofthebest‐fitlinefromtwodatapointsratherthantwopointsonthebest‐fitlineisacommonerror.Thiscanresultinasubstantialerrorintheslope.Usetwopointsonthebest‐fitlineforthisprocess,notdatapointstofindtheslopeofaline.Improperlyfindingtheslopeusingelementsofasinglecoordinatepoint.Whenfindingtheslopeofatangentline,studentssometimeswillmistakenlytakethecoordinatepointnearestwherethetangentlineisdrawnandfromthosecoordinatesattempttodeterminetheslope.Forinstance,at(x,y)=(5s,25m)thedatapointvaluesaremistakenlyusedtofindaslopeof25m/5s=5m/s.Failingtorecognizetrends.Graphsmustbeproperlyinterpretedthroughtheuseofgeneralizations(e.g.,doublethemass,andtheperiodincreasesby4)Improperlyinterpretingthephysicalmeaningoftheslope.Theunitsofaslopeareagiveawayasfarasinterpretingthephysicalmeaningofaslopeisconcerned.Theslopeisdefinedasthechangeintheyvaluedividedbyacorrespondingchangeinthexvalue.Hence,theunitsarethoseofthey‐axisdividedbythoseofthex‐axis.Ifdistance

(m)isonthey‐axis,andtime(s)onthex‐axis,thentheslope’sunitswillbem/swhichistheunitforspeed.Similarly,ifvelocity(m/s)isonthey‐axisandtime(s)isonthex‐axis,then(m/s)/s=m/s2,theunitofacceleration.Slopescanbeeitherpositiveornegative.Misinterpretingtheareaunderabest­fitlineorcurve.Tomoreeasilydeterminewhattheareaunderthecurverepresents,examinetheproductoftheunitsonthex‐andy‐axes.Forinstance,ifvelocityorspeed(m/s)isonthey‐axisandtime(s)isonthex‐axis,theproductoftheunitswillbemetersthatistheunitofdisplacementordistance.Hence,theareaunderthecurveofav‐tgraphisdisplacement.Similarly,theareaunderthecurveofana‐tgraphisvelocity.Notrecognizingerrorsindata.Whenonegraphsdata,largeerrorsinmeasurementusuallystandout.Theybecomenoticeablewhentheydeviatefromthetrendofthedata.Suchdatapointsshouldnotonlybequestioned,butthedatacollectionprocessforthisdatumrepeatedtoseeifthereisareadilyexplainableerror.Suspectdatashouldbestruckthroughratherthanerasedinthewrittenrecord.Updatethedatafileappropriately.Notunderstandingthemeaningofthebest­fitline.Abest‐fitlineisalinewhich,whendrawn,minimizesthesumofthesquaresofthedeviationsfromthelineusuallyintheydirection.Itisthelinethatmostcloselyapproximatesthetrendofthedata.

Preparing Graphs: Requirements for Introductory Physics Lab Activities

How NOT to Prepare a Graph

1. Pointprotectorsmissing.2. Connectingdatapointswithlines.3. Regressionlineshownwithdatapointline.4. Multiple(0,0)datapointsinattemptto“force”regressionlinethroughorigin.5. Nocolumnlabelsondataorgraph.6. Nounitsondataorgraph.7. Nolabelongraph.8. Labeldoesnotincludestudent’sname.

9. Providesalgebraicinterpretationofdata,notphysical(shownhereinNotesbox)10. Regressionformulamissingunitsonconstants.

Otherconcerns:

11. Failuretolinearizedatatogetastraight‐linerelationshipunlessotherwisecalledfor.12. Printinggraphwithoutdatatableonsamepage.13. Failuretoprintusealandscapeview.14. Wastingpaperandprintertoner.

How to Prepare a Graph

1. Pointprotectorsclearlyevident.2. Notconnectingdatapointswithlines.3. Regressionlineshownalone.4. No(0,0)datapoints;usesproportionalityforregressionline.5. Columnlabelsclearlyevidentondataandgraph.6. Unitsclearlyevidentondataorgraph.7. Labelongraph.8. Labelincludesstudent’sname.9. Providesphysicalinterpretationofdata,notalgebraic(showhereinNotessection).10. Regressionformulaincludesunitsonconstants.

Otherpointers:

11. Linearizedatatogetastraight‐linerelationshipunlessotherwisecalledfor.12. Printgraphwithdatatableonsamepage.13. Printusingalandscapeview,avoidusingportraitview.14. Wastingpaperandprintertoner.

Interpreting Slopes, Areas, and Intercepts of Graphs

Sometimesslopes,areas,andinterceptsingraphshavephysicalinterpretations.Atothertimestheyhavenorealmeaningatall.Considerhowonecaninterpretmeaningfultermsinthefollowinggraph.

Herewehavealinearrelationshipbetweenvelocityandtime.Theslopeofthegraphcanbefoundfromtherelationship

€

m =ΔyΔx

=y2 − y1x2 − x1

Becausethisisalinearrelationship,arbitrarilychoosingtwopairsof(x,y)coordinatesonthelineandperformingthecalculationresultsinaslopeof1m/s2.Thesloperepresentsthechangeinvelocitywithtimewhichisnothingotherthanacceleration.Notethatthecalculatedslopeunitsarethoseofacceleration.Whatdoestheareaunderthecurverepresent,say,fromtimeequals1stotimeequals5s?

Consideraddingupaseriesofverynarrowverticalcolumnstofindtotalarea.Theheightofeachverticalcolumnisvelocity,v,ataparticulartimeandthewidthofeachcolumnisΔt.TheproductofthesetermsisvΔtwhichequalsadistance.Hencethesumtotaloftheserectanglesgivingtheareaunderthecurveduringthespecifiedtimeintervalisthedistancethattheobjecthastraveledfromtime1sto5s.Canyouseehowthedistancetheobjecttravelsbetweent=1sandt=5sis32m?Youcanreadilyseethisbyaddingtheareaoftherectanglebelowandtotherightofthepoint(1,6)(area=heightxwidth=(6m/s)(4s)=24m)tothetriangletotheupperrightofthispoint(area=0.5heightxbase=0.5(4m/s)(4s)=8m).Notethe24m+8m=32m.They‐interceptcanalsobereadilyinterpretedintheabovegraph.Itismerelythevalueoftheytermattime=0s.Thatis,they‐interceptis5m/s,thevelocityoftheobjectatt=0s.Notallgraphsaresoreadilyinterpreted.Somegraph’sslopes,areas,andinterceptsmightwellbemeaningless.Considerthefollowingexamples,notallofwhicharemeaningful.Canyoutellwhichiswhich?Whatdoestheareaunderthecurveofanacceleration‐versus‐timegraphrepresent?Whatdoestheslopeofanacceleration‐versus‐timegraphrepresent?Whatdoesthey‐interceptofanacceleration‐versus‐timegraphrepresent?Whatdoestheareaunderthecurveofadisplacement‐versus‐timegraphrepresent?Whatdoestheslopeofadisplacement‐versus‐timegraphrepresent?Whatdoesthey‐interceptofadisplacement‐versus‐timegraphrepresent?

Physical Interpretations and Graphical Analysis

Creating realistic models from data requires more than a blind “best fit” strategy. A physical model needs to be created from an algebraic model if the new model is to have any useful meaning. For instance, let’s say that an experimenter measures the circumferences of a number of circular disks, as well as the diameters of these disks. There will be measurement error associated with each determination of circumference and diameter. A graph of circumference versus diameter produces the following result with the use of Graphical Analysis:

Note that an algebraic relationship results from a linear fit (y = mx + b) that is of the following form:

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y = 3.15x − 0.867mm

Identifying y with circumference, C, and x with the diameter, D, the equation translates to the following form:

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C = 3.15 ×D− 0.867mm This relationship suggests that if diameter equals zero, the circumference would have some negative value. This clearly is incorrect in our physical world. If D = 0, then C must equal 0. The way to resolve is problem is to create a physical model. Clearly, the relationship is linear, but the regression line must pass through the origin (0,0). A physical model can be created by forcing the regression line to pass through the origin. This is NOT done by including the data point 0,0; it IS done by using a different regression model, a proportionality (y = Ax), for the curve fit. Such a curve fit produces a different equation that we would call a physical model.

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C = 3.14D or more commonly

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C = πD = 2πr

where r is the radius of the circle. This physical model is not weighed down by the non-zero y intercept. When interpreting graphical results in lab, be certain to use an interpretation that corresponds to the physical world as we know it.

Report Writing

Hypothetico-deductive approaches to scientific experimentation

Scientific Values Scientists and engineers generally adhere to an informal set of values that serve as the basis of ethical conduct in the disciplines. Budding scientists and engineers are also expected to adhere to these rules of conduct. Unless these rules of conduct are expressly stated, students in an introductory lab course might never know them. What follows are one science educator author’s interpretation of expected personal behaviors. Cooperative attributes (and how students can demonstrate them in lab):

• Be well informed – Come to lab well prepared. This means not only to complete the required PreLab, but to have reviewed carefully the purpose of the indicated lab and given some thought to the required inquiry practices to be utilized in achieving stated goals. Read pertinent sections of the Student Laboratory Handbook.

• Stay focused – When in the laboratory setting, the emphasis should be on the first part of the word laboratory (labor) and not on the second (oratory). Work diligently on the task at hand, and see it through to completion with dispatch. Everyone should contribute to the best of his or her ability.

• Evaluate alternatives – Listen to others in your group; perhaps someone with an alternative approach might, in fact, be pointing out the best approach. Consider all arguments carefully before making a decision.

• Take supportable positions – Positions in lab should be based on evidence and critical thinking. If you hold a position contrary to that of you colleagues, justify it.

• Being sensitive to others’ positions – Not everyone will agree in lab. If you disagree with another person in your group, then find a solution using reason and evidence. Criticize the idea, not the person.

• Seek precision – Don’t be satisfied with “good enough.” Try your best to minimize experimental error and improve your results. Strive for accuracy in word and action.

• Proceed in a logical and orderly manner – For many of you, this will be the first time you have been asked to design and conduct experiments. Think the process through carefully before collecting data.

Individual attributes (and how students can demonstrate them in lab):

• Be well informed – Come to lab prepared with an understanding of what it is you intend to accomplish and ideas for how you will accomplish it. This generally means completing the PreLab work, and reading through the lab guidelines in their entirety while attempting to understand. Also, don’t forget to read the articles in the Student Laboratory Handbook; it has many good articles dealing with practical lab matters.

• Stay focused – While working in lab, it is important to stay focused on the task at hand. Avoid wayward discussions. The focus in laboratory should be on labor and not on oratory.

• Be willing to evaluate alternatives – While you will arrive in lab with your own ideas about how a task should be accomplished, carefully weigh ideas and comments by your lab partner(s) dealing with alternative approaches before coming to a firm decision.

• Maintain only supportable positions – You should always be sensitive to the positions of other. If, however, there is a justifiable reason for taking one approach over another, then stand your ground. Supportable positions are tenable. Ask for outside commentary from your lab instructor if you have reached an intractable disagreement about how to proceed.

• Seek precision – Always strive to be as accurate as would reasonably be expected of a scientist. There is no excuse for sloppy lab work. It would never be accepted by the scientific community in professional research, and won’t be accepted in the introductory lab setting as well.

• Proceed in a logical and orderly manner – Plan your work before executing it. Failure to do so is like using a calculator to “solve a problem.” Calculators don’t solve problems; people do. The calculator literally calculates a solution, and can only do so after a human has reasoned out the approach to a solution. Merely using scientific hardware and software won’t solve problems; nothing can replace good intellectual thought.

• Be sensitive to others– When working with your colleague(s), be considerate of his or her needs and concerns. Show proper respect; listen to others; think before you act.

• Think critically – There are many definitions of critical thinking, and any of the following would apply: “reasonable reflective thinking that is focused on deciding what to do and what to believe” OR “interpreting, analyzing or evaluating information, arguments or experiences with a set of reflective attitudes, skills, and abilities to guide our thoughts, beliefs and actions” OR “examining the thinking of others to improve our own.” Strive to apply critical thinking to your tasks.

• Be intellectually honest – Evaluate all pertinent evidence carefully and systematically. Avoid prejudicial decision-making.

• Avoid unwarranted closure – Avoid leaping to conclusions if data don’t support a particular position. • Demonstrate personal integrity – Never “fudge” data to make it fit the expected outcomes. Never eliminate

data unless there is a clear and defensible reason.